Diffusion-like solutions of the Schrodinger equation for a time-dependent potential well

1992 ◽  
Vol 25 (1) ◽  
pp. 241-249 ◽  
Author(s):  
A Devoto ◽  
B Pomorisac
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Time Dependent ◽  
Potential Well ◽  
Dependent Potential

Time-dependent versus time-independent probabilities of transmission and reflection of a quantum particle incident upon a step potential and a square potential well

2016 ◽  
Vol 94 (1) ◽  
pp. 9-14 ◽  
Author(s):  
Mark R.A. Shegelski ◽  
Kevin Malmgren ◽  
Logan Salayka-Ladouceur
Keyword(s):  
Quantum Mechanics ◽  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Quantum Particle ◽  
Time Dependent ◽  
Potential Well ◽  
Step Potential ◽  
Exact Analytical Expression ◽  
Square Well ◽  
Transmission And Reflection

We investigate the transmission and reflection of a quantum particle incident upon a step potential decrease and a square well. The probabilities of transmission and reflection using the time-independent Schrödinger equation and also the time-dependent Schrödinger equation are in excellent agreement. We explain why the probabilities agree so well. In doing so, we make use of an exact analytical expression for the square well for time-dependent transmission and reflection, which reveals additional interesting and unexpected results. One such result is that transmission of a wave packet can occur with the probability of transmission depending weakly on the initial spread of the packet. The explanations and the additional results will be of interest to instructors of and students in upper year undergraduate quantum mechanics courses.

scholarly journals Efficient methods for linear Schrödinger equation in the semiclassical regime with time-dependent potential

2016 ◽  
Vol 472 (2193) ◽  
pp. 20150733 ◽  
Author(s):  
Philipp Bader ◽  
Arieh Iserles ◽  
Karolina Kropielnicka ◽  
Pranav Singh
Keyword(s):  
Schrödinger Equation ◽  
Numerical Experiments ◽  
Schrodinger Equation ◽  
Linear Time ◽  
Time Dependent ◽  
Magnus Expansion ◽  
Link Type ◽  
Dependent Potential ◽  
Semiclassical Regime

We build efficient and unitary (hence stable) methods for the solution of the linear time-dependent Schrödinger equation with explicitly time-dependent potentials in a semiclassical regime. The Magnus–Zassenhaus schemes presented here are based on a combination of the Zassenhaus decomposition (Bader et al. 2014 Found. Comput. Math. 14 , 689–720. ( doi:10.1007/s10208-013-9182-8 )) with the Magnus expansion of the time-dependent Hamiltonian. We conclude with numerical experiments.

The symmetry group of the quantum harmonic oscillator in an electric field

Open Physics ◽  
2008 ◽  
Vol 6 (3) ◽  
Author(s):  
Juan Lejarreta ◽  
Jose Cerveró
Keyword(s):  
Schrödinger Equation ◽  
Electric Field ◽  
Harmonic Oscillator ◽  
Schrodinger Equation ◽  
Group Structure ◽  
Time Dependent ◽  
Frequency Function ◽  
General Group ◽  
Dependent Potential ◽  
The Relationship

AbstractIn this paper we present two results. First, we derive the most general group of infinitesimal transformations for the Schrödinger Equation of the general time-dependent Harmonic Oscillator in an electric field. The infinitesimal generators and the commutation rules of this group are presented and the group structure is identified. From here it is easy to construct a set of unitary operators that transform the general Hamiltonian to a much simpler form. The relationship between squeezing and dynamical symmetries is also stressed. The second result concerns the application of these group transformations to obtain solutions of the Schrödinger equation in a time-dependent potential. These solutions are believed to be useful for describing particles confined in boxes with moving boundaries. The motion of the walls is indeed governed by the time-dependent frequency function. The applications of these results to non-rigid quantum dots and tunnelling through fluctuating barriers is also discussed, both in the presence and in the absence of a time-dependent electric field. The differences and similarities between both cases are pointed out.

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